Knowledge is Wealth; Increases by Sharing

Month: July 2011

A constant struggle for power is integral part of human nature. Whether a community is managed by the democracy, a monarch, or a dictator, there is an ongoing power struggle. In a democracy political parties contest to control the parliament, while a monarch or a dictator are themselves the center of power. It is a constant struggle to remain in power by influencing the minds of the citizens. Such phenomenon has been observed since ages and is never ending. We can witness such power struggles in almost everything—from the financial institutions effecting lives of millions, or few companies controlling the computer industry, and the internet market. The 21st century is about the dominance over the Web – World Wide Web.

The British East India Company, whose sole purpose was trading, came to India in early seventeenth century in search of species. Their initial intentions were nothing extraordinary, but trading. Then why British stayed and ruled India for over 300 years. The greedy nature of human beings is not the only one to blame for. There are rules called ‘winner’s takes-it-all’ and ‘rich-get-richer’. In that century, British were not the only empire competing for dominance, but there were French, Dutch, and Portuguese, also. But among these competitors the British had better network, resources, and skills to expand their network, hence, they succeeded. If it were not British then it might have been someone else or India’s own emperors exploiting India, because at the end winner takes-it-all.

Similar future is seen for the Internet; currently, small number of technology companies is controlling more than eighty percent of the market. The 80/20 rule comes into picture. Microsoft controls more than eighty percent of operating system market. It is always that these big players take a greater chunk of market, leaving others behind. These bigger corporations has bigger network. Until recently, most usage of computers and software has been limited to offices and had limited effects on our lives. But these technologies led to the birth of the internet, which already started affecting our lives.

It can, already, be seen by our dependence on search engines. The search engines have become integral part of lives. The world “Google” has become part of the language—don’t know how to cook? Let us google it. It can be argued that popularity of these search engines is due to the belief that they provide with most relevant search results. It is true up to some extent. But the extensive popularity of these search engines had triggered various phenomenons due to which the recommendations are biased. Let’s go back to “rich-get-richer”. Search engines take into consideration number links pointing to a webpage as a key criteria in determining the rank of a webpage. The Pages with more number of links are ranked higher and have better chances of appearing in top results. Moreover, people have tendency to view top results and ignoring the rest. Due to this, already rich page now have even more chances of getting new links; subsequently become even more popular with the search engines. Since we trust the search engines, it affects us. This situation has risen recently due to increase in use of search engines. The results we see are inherently biased, and not necessarily provide right content. Note that, such phenomenon is not intended by the developers of these algorithms, but it’s an occurrence which they might not have visualized.

Technology has started affecting the lives of people in the manner which was never envisioned. On top of it, dominance of these corporations is increasing. Issues such as data and personal privacy, net neutrality, etc. are prevalent. Moreover, with the spread of cloud computing, we are about to lose control over our own data. Several questions need to be answered. How much we can trust these multi-million dollar corporations? How much are we willing to let go? Will there be a British East India Company with right network, resources, and skills to expand their network, and eventually, dominate the internet?

Nevertheless, war has begun; battleground is the Web and target is the common man.

The core of the fuzzy logic is the computational backbone of the fuzzy arithmetic. Power of the fuzzy logic is due to the strong principles governing the computation of the fuzzy sets. In this note we present fundamental computational rules of fuzzy logic.

Fuzzy Set

Given a universal set X, a fuzzy set A is described by a membership function mapping for every element of X to the unit interval [0 1]:

In general, a fuzzy set is expressed as a pair {x, µA}, where x is an element of universal set X and µA is its degree of membership in set A.

Other convenient notation of fuzzy set is

Basic Definitions of Fuzzy Set

Fuzzy Relation

An n-ary fuzzy relation consist of set of ordered pairs of elements (x1, x2,…, xn) of the universal product set (X1 × X2 ×…×Xn) with its degree of membership µR(x1, x2,…, xn).

Height

The height of fuzzy set A is the supremum (maximum) of the membership function µA.

For a fuzzy relation R,

Normal Fuzzy Set

If hgt(A) = 1, then fuzzy set A is normal;

Sub Normal Fuzzy Set

if hgt(A) < 1, then fuzzy set A is subnormal.

Normalization

Normalization if process of converting a subnormal fuzzy set in to a normal fuzzy set.

Core

The core of a fuzzy set A is the crisp set of all the elements with membership degree equal to 1.

For fuzzy relation R,

Support

The support of a fuzzy set A is the crisp set of all the elements with membership degree greater than zero.

For fuzzy relation R,

α-cut

The alpha cut of a fuzzy set A is the crisp set of all the elements with membership degree greater than or equal to a threshold .

Also,

is the strong α-cut of fuzzy set A. Similarly, α-cuts for a fuzzy relation R can be defined.